Lennon appears set to follow as Sean Dyche's seventh-placed side prepare to face Manchester United this weekend.
He has made close to 80 appearances for Everton, scoring nine goals and, following psychiatric treatment last spring, he was brought back into the first-team fold earlier this season after falling out of favour last term.
The reports said that Lennon has agreed personal terms with Burnley but is yet to undergo a medical.
He has since made a further 19 appearances this season, but is now thought to be considered surplus to requirements following the £20m arrival of Theo Walcott from Arsenal, and the recent return from injury of Yannick Bolasie.
The England worldwide recovered from a stress-related illness in pre-season, having missed the last three months of the previous campaign, in which he made just 13 appearances.
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